EDHEC-Risk Institute October 2016

Multi-Dimensional Risk and Performance Analysis for Equity Portfolios — October 2016

3. Applications of Fundamental Beta

expected return of the portfolio p , conditional on the stocks’ weights (actually sector allocation) at date t in the portfolio:

(3.1)

These quantities are called a “fundamental alpha” and a “fundamental beta”. For N stocks, the model has 6 N +2 S coefficients (with S sectors) to estimate. Each stock is attributed to only one sector which is represented with a dummy variable. Sector coefficients θ α ,0,s ( i ) and θ β ,0,s ( i ) are common to stocks that belong to the same sector. The coefficients are not independent from one stock to the other because stocks from a given sector share the same parameters θ α ,0,s ( i ) and θ β ,0,s ( i ) . The model is estimated by minimising the sum of squared residuals ε i,t over all dates and stocks. minimise

(3.2)

The constant market premium in this equation must be replaced by the conditional market premium in case this parameter is time-varying.

The conditional beta of a portfolio is the weighted sum of those of the constituents:

Since each stock belongs to one sector only, we have:

(3.3)

In order to include the country attribute in this decomposition, one needs a version of the fundamental beta and alpha in which these parameters depend on the

where denotes the j -th sector weight in portfolio p . We can decompose the

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